International Electronic Journal of Mathematics Education

International Electronic Journal of Mathematics Education Indexed in ESCI
Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers
In-text citation: (Rizos & Adam, 2022)
Reference: Rizos, I., & Adam, M. (2022). Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), em0686.
In-text citation: (1), (2), (3), etc.
Reference: Rizos I, Adam M. Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. INT ELECT J MATH ED. 2022;17(3), em0686.
In-text citation: (Rizos and Adam, 2022)
Reference: Rizos, Ioannis, and Maria Adam. "Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers". International Electronic Journal of Mathematics Education 2022 17 no. 3 (2022): em0686.
In-text citation: (Rizos and Adam, 2022)
Reference: Rizos, I., and Adam, M. (2022). Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), em0686.
In-text citation: (Rizos and Adam, 2022)
Reference: Rizos, Ioannis et al. "Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers". International Electronic Journal of Mathematics Education, vol. 17, no. 3, 2022, em0686.
In-text citation: (1), (2), (3), etc.
Reference: Rizos I, Adam M. Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. INT ELECT J MATH ED. 2022;17(3):em0686.


In this paper, we present parts of an online qualitative research project which lasted a whole semester about the understanding of real numbers by sixty first-year Greek mathematics undergraduate students during the COVID-19 pandemic. We focus on the distinction between rational and irrational numbers and their location on the number line using straightedge and compass. The results, came of questionnaires, focused interviews and tasks, disclosed some gaps regarding real numbers that the students had from school and revealed confusion between irrational numbers and their decimal approximation. The results also led us to group students’ conceptions of rational and irrational numbers into five categories related to the number and pattern of decimal digits. Teaching practices and perspectives as well as the, negative, impact of the pandemic on teaching and learning mathematics are discussed.


Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).


  • Adhikari, K. P. (2021). Difficulties and misconceptions of students in learning limit. Interdisciplinary Research in Education, 5(1-2), 15-26.
  • Adiredja, A. P. (2021). The pancake story and the epsilon-delta definition. PRIMUS, 31(6), 662-677.
  • Alibert, D., & Thomas, M. (2002). Research on mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking (pp. 215-230). Springer.
  • Apostol, T. M. (1997). Modular functions and dirichlet series in number theory. Springer.
  • Arcavi, A., Bruckheimer, M., & Ben-Zvi, R. (1987). History of mathematics for teachers: The case of irrational numbers. For the Learning of Mathematics, 7(2), 18-23.
  • Artigue, M. (1997). Teaching and learning elementary analysis: What can we learn from didactical research and curriculum evolution? In G. Makrides (Ed.), Proceedings of 1st Mediterranean Conference on Mathematics (pp. 207-219).
  • Bampili, A. C., Zachariades, T., & Sakonidis, C. (2019). The transition from high school to university mathematics: The effect of institutional issues on students’ initiation into a new practice of studying mathematics. In Proceedings of the 11th Congress of the European Society for Research in Mathematics Education. Utrecht University.
  • Bansilal, S., & Mkhwanazi, T. W. (2021). Pre-service student teachers’ conceptions of the notion of limit. International Journal of Mathematical Education in Science and Technology.
  • Biza, I., Christou, C., & Zachariades, T. (2008). Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean geometry to analysis. Research in Mathematics Education, 10(1), 53-70.
  • Borba, M. C. (2021). The future of mathematics education since COVID-19: Humans-with-media or humans-with-non-living-things. Educational Studies in Mathematics, 108, 355-400.
  • Burgess, R. G. (1984). In the field: An introduction to field research. Routledge.
  • Chinnappan, M., & Forrester, T. (2014). Generating procedural and conceptual knowledge of fractions by pre-service teachers. Mathematics Education Research Journal, 26(4), 871-896.
  • Christou, K. P., & Vamvakoussi, X. (2021). Natural number bias on evaluations of the effect of multiplication and division: The role of the type of numbers. Mathematics Education Research Journal.
  • Chrystal, G. (1889). Algebra, An elementary text-book, part I. A. & C. Black.
  • Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Springer.
  • Denzin, N. K. (2009). The research act: A theoretical introduction to sociological methods. Transaction Publishers.
  • Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational numbers in high-school students and prospective teachers. Educational Studies in Mathematics, 29, 29-44.
  • Fried, N. M. (2001). Can mathematics education and history of mathematics coexist? Science & Education, 10(4), 391-408.
  • Furinghetti, F. (2020). Rethinking history and epistemology in mathematics education. International Journal of Mathematical Education in Science and Technology, 51(6), 967-994.
  • Gellert, U., & Jablonka, E. (2009). The demathematising effect of technology. In P. Ernest, B. Greer, & B. Sriraman (Eds), Critical Issues in Mathematics Education (pp. 19-24). IAP.
  • Giannakoulias, E., Souyoul, A., & Zachariades, T. (2007). Students’ thinking about fundamental real numbers properties. In Proceedings of the 5th Congress of the European society for Research in Mathematics Education (pp. 416-425).
  • Greenwald, S. J. (2016). Popular culture in teaching, scholarship, and outreach: The Simpsons and Futurama. In J. Dewar, P. Hsu, & H. Pollatsek (Eds), Mathematics education (pp. 349-362). Springer.
  • Gulikers, I., & Blom, K. (2001). “A historical angle”, A survey of recent liter­ature on the use and value of history in geometrical education. Educa­tional Studies in Mathematics, 47, 223-258.
  • Guven, B., Cekmez, E., & Karatas, I. (2011). Examining preservice elementary mathematics teachers’ understandings about irrational numbers. PRIMUS, 21(5), 401-416.
  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23.
  • Helfgott, M. (2004). Two examples from the natural sciences and their relationship to the history and pedagogy of mathematics. Mediterranean Journal for Research in Mathematics Education, 3(1-2), 147-164.
  • Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A., da Silva, C.M.S., & Weeks, C. (2000). The use of original sources in the mathematics classroom. In J. Fauvel, & J. van Maanen (Eds), History in mathematics education: The ICMI study (pp. 291-328). Kluwer Academic Publishers.
  • Jankvist, U. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education. Educational Studies in Mathemat­ics, 71, 235-261.
  • Karalis, T., & Raikou, N. (2020). Teaching at the times of COVID-19: Inferences and implications for higher education pedagogy. International Journal of Academic Research in Business and Social Sciences, 10(5), 479-493.
  • Katz, V. J. (1993). Using the history of calculus to teach calculus. Science & Education, 2, 243-249.
  • Keitel, C. (1989). Mathematics education and technol­ogy. For the Learning of Mathematics, 9(1), 7-13.
  • Kidron, I. (2016). Understanding irrational numbers by means of their representation as non-repeating decimals [Paper presentation]. 1st Conference of International Network for Didactic Research in University Mathematics, Montpellier, France.
  • Klein, M. (1972). Mathematical thought from ancient to modern times, vol. 1-3. Oxford University Press.
  • Kleiner, I. (2001). History of the infinitely small and the infinitely large in calculus. Educational Studies in Mathematics, 48, 137-174.
  • Mamona, J. (1987). Students’ interpretations of some concepts of mathematical analysis [Unpublished PhD thesis]. University of Southampton.
  • Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M. Limón, & L. Mason (Eds), Reconsidering conceptual change: Issues in theory and practice (pp. 233-258). Kluwer Academic Publishers.
  • Merton, R. K., & Kendall, P. L. (1946). The focused interview. The American Journal of Sociology, 51(6), 541-557.
  • Mishra, L., Gupta, T., & Shree, A. (2020). Online teaching-learning in higher education during lockdown period of COVID-19 pandemic. International Journal of Educational Research Open, 1, 100012.
  • Moseley, B. (2005). Students’ early mathematical representation knowledge: The effects of emphasizing single or multiple perspectives of the rational number domain in problem solving. Educational Studies in Mathematics, 60, 37-69.
  • Moskal, B. M. & Magone, M. E. (2000). Making sense of what students know: Examining the referents, relationships and modes students displayed in response to a decimal task. Educational Studies in Mathematics, 43(3), 313-335.
  • Mulenga, E. M., & Marbán, J. M. (2020). Is COVID-19 the gateway for digital learning in mathematics education? Contemporary Educational Technology, 12(2), ep269.
  • NCTM. (2006). Navigating through number and operations in grades 9-12. National Council of Teachers of Mathematics.
  • Neugebauer, O. (1957). The exact sciences in antiquity. Brown University Press.
  • O’ Connor, M. C. (2001). Can any fraction be turned into a decimal? A case study of a mathematical group discussion. Educational Studies in Mathematics, 46, 143-185.
  • Pantsar, M. (2016). The great gibberish–Mathematics in western popular culture. In B. Larvor (Ed.), Mathematical cultures: The London meetings 2012-2016 (pp. 409-437). Springer.
  • Patel, P., & Varma, S. (2018). How the abstract becomes concrete: Irrational numbers are understood relative to natural numbers and perfect squares. Cognitive Science, 42(5), 1-35.
  • Patronis, T., & Thomaidis, Y. (1997). On the arithmetization of school geometry in the setting of modern axiomatics. Science & Education, 6, 273-290.
  • Peled, I., & Hershkovitz, S. (1999). Difficulties in knowledge integration: Revisiting Zeno’s paradox with irrational numbers. International Journal of Mathematical Education in Science and Technology, 30(1), 39-46.
  • Pitta-Pantazi, D., Christou, C., & Pittalis, M. (2020). Number teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer.
  • Polya, G. (1945). How to solve it. Princeton University Press.
  • Potari, D., Psycharis, G., Sakonidis, C., & Zachariades, T. (2019). Collaborative design of a reform-oriented mathematics curriculum: Contradictions and boundaries across teaching, research, and policy. Educational Studies in Mathematics, 102, 417-434.
  • Rizos, I., & Gkrekas, N. (2022). Teaching and learning sciences within the COVID-19 pandemic era in a Greek university department. U. Porto Journal of Engineering, 8(1), 73-83.
  • Rizos, I., Patronis, A., & Lappas, D. (2017). “There is one geometry and in each case there is a different formula”. Students’ conceptions and strategies in an attempt of producing a Minkowskian metric on space-time. Science & Education, 26(6), 691-710.
  • Rizos, I., Patronis, T., & Papadopoulou, A. (2021). Difficulties in basic arithmetic and geometry as related to school algebra and the current effect of demathematization. For the Learning of Mathematics, 41(1), 37-39.
  • Roell, M., Viarouge, A., Houdé, O., & Borst, G. (2017). Inhibitory control and decimal number comparison in school-aged children. PLoS ONE, 12(11), e0188276. pone.0188276
  • Roell, M., Viarouge, A., Houdé, O., & Borst, G. (2019). Inhibition of the whole number bias in decimal number comparison: A developmental negative priming study. Journal of Experimental Child Psychology, 177, 240-247.
  • Roh, K. H. (2008). Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69(3), 217-233.
  • Siegler, R. S., & Lortie-Forgues, H. (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science, 26(4), 346-351.
  • Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994-2004.
  • Sierpinska, A. (1989). On 15-17 years old students’ conceptions of functions, iteration of functions and attractive fixed points. Académie des Sciences de Pologne [Polish Academy of Sciences].
  • Sierpinska, A. (2005). On practical and theoretical thinking and other false dichotomies in mathematics education. In M. Hoffmann, J. Lenhard, & F. Seeger (Eds), Activity and sign: Grounding mathematics education (pp. 117-135). Springer.
  • Sirotic, N., & Zazkis, R. (2007). Irrational numbers on the number line–where are they? International Journal of Mathematical Education in Science and Technology, 38(4), 477-488.
  • Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Kluwer Academic Publishers.
  • Smith, C. L., Solomon, G. E. A., & Carey, S. (2005). Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology, 51, 101-140.
  • Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503-518.
  • Stewart, D. W., & Shamdasani, P. N. (1990). Focus group: Theory and practice. SAGE.
  • Tall, D. (1992). Students’ difficulties in calculus. In Proceedings of Working Group 3 on Students’ Difficulties in Calculus (pp. 13-28).
  • Tall, D. (2013). How humans learn to think mathematically. Exploring the three worlds of mathematics. Cambridge University Press.
  • Tian, J., & Siegler, R. S. (2018). Which type of rational numbers should students learn first? Educational Psychology Review, 30(2), 351-372.
  • Tirosh, D., Fischbein, E., Graeber, A. O., & Wilson, J. W. (1998). Prospective elementary teachers’ conceptions of rational numbers.
  • Tzanakis, C., Arcavi, A., Correia de Sa, C., Isoda, M., Lit, C. K., Niss, M., de Carvalho, J. P., Rodriguez, M., & Siu M. K. (2000). In­tegrating history of mathematics in the classroom: An analytic survey. In J. Fauvel, & J. van Maanen (Eds), History in mathematics education: The ICMI study (pp. 201-240). Kluwer Academic Publishers.
  • Tzekaki, Μ., Sakonidis, C., & Kaldrimidou, M. (2003). Mathematics education in Greece: A study. In A. Gagatsis, & S. Papastavridis (Eds), Proceedings of the 3rd Mediterranean Conference on Mathematical Education (pp. 629-639). Hellenic Mathematical Society, Cyprus Mathematical Society.
  • Uegatani, Y., Nakawa, N., & Kosaka, M. (2021). Changes to tenth-grade Japanese students’ identities in mathematics learning during the COVID-19 pandemic. International Electronic Journal of Mathematics Education, 16(2), em0638.
  • Valiron, G. (1971). The origin and evolution of the notion of an analytic function of one variable. In F. Le Lionnais (Ed.), Great currents of mathematical thought (pp. 156-173). Dover.
  • Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453-467.
  • Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers are there in a rational number interval? Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds), Reframing the conceptual change approach in learning and instruction (pp. 265-282). Elsevier.
  • van Hiele, P. M. (1957). De problematiek van het inzicht [The problem of insight] [Unpublished thesis]. Utrecht University.
  • van Hiele, P. M., & van Hiele-Geldof, D. (1958). A method of initiation into geometry at secondary schools. In H. Freudenthal (Ed.), Report on methods of initiation into geometry. J. B. Wolters.
  • van Hoof, J., Lijnen, T., Verschaffel, L., & van Dooren, W. (2013). Are secondary school students still hampered by the natural number bias? A reaction time study on fraction comparison tasks. Research in Mathematics Education, 15(2), 154-164.
  • Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Springer.
  • von Fritz, K. (1945). The discovery of incommensurability by Hippasus of Metapontum. Annals of Mathematics, 46(2), 242-264.
  • Voskoglou, M. G., & Kosyvas, G. D. (2012). Analyzing students’ difficulties in understanding real numbers. Journal of Research in Mathematics Education, 1(3), 301-336.
  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101-119.
  • Williams, E. R. (1979). An investigation of senior high school students’ understanding of mathematical proof [PhD thesis, University of Alberta].
  • Zachariades, T., Christou, C., & Pitta-Pantazi, D. (2013). Reflective, systemic and analytic thinking in real numbers. Educational Studies in Mathematics, 82, 5-22.
  • Zazkis, R., & Sirotic, N. (2004). Making sense of irrational numbers: Focusing on representation. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 497-504).
  • Zazkis, R., & Sirotic, N. (2010). Representing and defining irrational numbers: Exposing the missing link. CBMS Issues in Mathematics Education, 16, 1-27.


This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.